Even in 2*3, the 3 refers to the number of objects, while the 2 refers tothe number of groups of 3. So the interpretations of the two numbers arenecessarily different even with whole numbers. When considering (-2)*(-3), the -3 refers to a directed amount, and the-2 refers to a "directed number" of groups of -3. (Refer to my excitingpost from yesterday about the mathematical basis for integers, and howinteger multiplication is not the same as whole number multiplication.) I continue to argue that the conceptual basis for this is non-trivial,and many of us (myself included) have rather tentative understandings ofinteger multiplication. Which is why we have a tough time teaching it. Gary

At 3:10 PM 4/3/96, Lou Talman wrote:>Tim hendrix wrote:>>> An important part of the Postman approach or any other story>> approach to unearth the concept of off-setting properties of multiplying>> two negatives is that each integer in the product must have a different>> interpretation:>>This observation is itself important; it ties in nicely with my observation>about the way we overload the "-" sign in our discussions of arithmetic with>signed numbers. The fact that we must give different interpretations to the>"-" signs when we give plausibility arguments to justify (-)*(-) = (+) is an>indication of the fundamental artificiality of those plausibility arguments.>Notice that in my earlier post I *did not* overload the "-" sign. I used "-">as the unary negation operator *only*. That's one of the reasons I say that>this is the "real" reason why (-)*(-) = (+).>>--Lou Talman

W. Gary MartinCurriculum Research and Development GroupUniversity of Hawaii